2.11  Spin-Orbit Coupling

The rotation of planets and natural satellites is affected by the gravitational forces from other celestial bodies. As an extended application of the Lagrangian method for forced rigid bodies, we consider the rotation of celestial objects subject to gravitational forces.

We first develop the form of the potential energy for the gravitational interaction of an extended body with an external point mass. With this potential energy and the usual rigid-body kinetic energy we can form Lagrangians that model a number of systems. We will take an initial look at the rotation of the Moon and Mercury; later, after we have developed more tools, we will return to study these systems.

2.11.1  Development of the Potential Energy

The first task is to develop convenient expressions for the gravitational potential energy of the interaction of a rigid body with a distant mass point. A rigid body can be thought of as made of a large number of mass elements, subject to rigid coordinate constraints. We have seen that the kinetic energy of a rigid body is conveniently expressed in terms of the moments of inertia of the body and the angular velocity vector, which in turn can be represented in terms of a suitable set of generalized coordinates. The potential energy can be developed in a similar manner. We first represent the potential energy in terms of moments of the mass distribution and later introduce generalized coordinates as particular parameters of the potential energy.

The gravitational potential energy of a mass point and a rigid body (see figure 2.9) is the sum of the potential energy of the mass point with each mass element of the body:

where M' is the mass of the external point mass, r is the distance between the point mass and the constituent mass element with index , m is the mass of this constituent element, and G is the gravitational constant. Let R be the distance of the center of mass of the rigid body from the point mass; R is the magnitude of the vector - , where the external mass point has position and the center of mass of the rigid body has position . The vector from the center of mass to the constituent with index is and has magnitude . The distance r is then given by the law of cosines as r² = R² + ² - 2 R cos , where is the angle between - and . The potential energy is then

This is complete, but we need to find a representation that does not mention each constituent.

Typically, the size of celestial bodies is small compared to the distance between them. We can make use of this to find a more compact representation of the potential energy. If we expand the potential energy in the small ratio /R we find

where P_l is the lth Legendre polynomial.¹⁵ Interchanging the order of the summations yields:

Successive terms in this expansion of the potential energy typically decrease very rapidly because celestial bodies are small compared to the distance between them. We can compute an upper bound to the size of these terms by replacing each factor in the sum over by an upper bound. The Legendre polynomials all have magnitudes less than one for arguments in the range - 1 to 1. The distances are all less than some maximum extent of the body _max. The sum over m times these upper bounds is just the total mass M times the upper bounds. Thus

We see that the upper bound on successive terms decreases by a factor _max / R. Successive terms may be smaller still. For large bodies the gravitational force is strong enough to overcome the internal material strength of the body, so the body, over time, becomes nearly spherical. Successive terms in the expansion of the potential are measures of the deviation of the mass distribution from a spherical mass distribution. Thus for large bodies the higher-order terms are small because the bodies are nearly spherical.

Consider the first few terms in l. For l = 0 the sum over just gives the total mass M of the rigid body. For l = 1 the sum over is zero, as a consequence of choosing the origin of the to be the center of mass. For l = 2 we have to do a little more work. The sum involves second moments of the mass distribution, and can be written in terms of moments of inertia of the rigid body:

where A, B, and C are the principal moments of inertia, and I is the moment of inertia of the rigid body about the line between the center of mass of the body to the external point mass. The moment I depends on the orientation of the rigid body relative to the line between the bodies. The contributions to the potential energy up to l = 2 are then¹⁶

Let = cos _a, ß = cos _b, and = cos _c be the direction cosines of the angles _a, _b and _c between the principal axes hata, , and and the line between the center of mass and the point mass.¹⁷ A little algebra shows that I = ² A + ß² B + ² C. The potential energy is then

This is a good first approximation to the potential energy of interaction for most situations in the solar system; if we intended to land on the Moon we probably would want to take into account higher-order terms in the expansion.

Exercise 2.14.  

a.  Fill in the details that show that the sum over constituents in equation (2.72) can be expressed as written in terms of moments of inertia. In particular, show that

and that

b.  Show that if the principal moments of inertia of a rigid body are A, B, and C, then the moment of inertia about an axis that goes through the center of mass of the body with direction cosines , ß, and relative to the principal axes is

2.11.2  Rotation of the Moon and Hyperion

The approximation to the potential energy that we have derived can be used for a number of different problems. It can be used to investigate the effect of oblateness on the evolution of an artificial satellite about the Earth, or to incorporate the effect of planetary oblateness on the evolution of the orbits of natural satellites, such as the Moon or the Galilean satellites of Jupiter. However, as the principal application here, we will use it to investigate the rotational dynamics of natural satellites and planets.

The potential energy depends on the position of the point mass relative to the rigid body and on the orientation of the rigid body. Thus the changing orientation is coupled to the orbital evolution; each affects the other. However, in many situations the effect of the orientation of the body on the evolution of the orbit may be ignored. One way to see this is to look at the relative magnitudes of the two terms in the potential energy (2.74). We already know that the second term is guaranteed to be smaller than the first by a factor of (_max / R)², but often it is much smaller still because the body involved is nearly spherical. For example, the radius of the Moon is about a third the radius of the Earth and the distance to the Moon is about 60 Earth-radii. So the second term is smaller than the first by a factor of order 10^-4 due to the size factors. In addition, the Moon is roughly spherical and for any orientation the combination A + B + C - 3I is of order 10^-4C. Now C is itself of order (2/5)MR², because the density of the Moon does not vary strongly with radius. So for the Moon the second term is of order 10^-8 relative to the first. Even radical changes in the orientation of the Moon would have little dynamical effect on its orbit.

We can learn some important qualitative aspects of the orientation dynamics by studying a simplified model problem. First, we assume that the body is rotating about its largest moment of inertia. This is a natural assumption. Remember that for a free rigid body the loss of energy while conserving angular momentum leads to rotation about the largest moment of inertia. This is observed for most bodies in the solar system. Next, we assume that the spin axis is perpendicular to the orbital motion. This is a good approximation for the rotation of natural satellites, and is a natural consequence of tidal friction -- dissipative solid-body tides raised on the satellite by the gravitational interaction with the planet. Finally, for simplicity we take the rigid body to be moving on a fixed elliptic orbit. This may approximate the motion of some physical systems, provided the time scale of the evolution of the orbit is large compared to any time scale associated with the rotational dynamics we are investigating. So we have a nice toy problem, one that has been used to investigate the rotational dynamics of Mercury, the Moon, and other natural satellites. It makes specific predictions concerning the rotation of Phobos, a satellite of Mars, that can be compared with observations. It provides a basic explanation of the fact that Mercury rotates precisely three times for every two orbits it completes, and is the starting point for understanding the chaotic tumbling of Saturn's satellite Hyperion.

We are assuming that the orbit does not change or precess. The orbit is an ellipse with the point mass at a focus of the ellipse. The angle f (see figure 2.10) measures the position of the rigid body in its orbit relative to the point in the orbit at which the two bodies are closest.¹⁸ We assume the orbit is a fixed ellipse, so the angle f and the distance R are periodic functions of time, with period equal to the orbit period. With the spin axis constrained to be perpendicular to the orbit plane, the orientation of the rigid body is specified by a single degree of freedom: the orientation of the body about the spin axis. We specify this orientation by the generalized coordinate that measures the angle to the hata principal axis from the same line from which we measure f, the line through the point of closest approach.

Having specified the coordinate system, we can work out the details of the kinetic and potential energies, and thus find the Lagrangian. The kinetic energy is

where C is the moment of inertia about the spin axis and the angular velocity of the body about the axis is . There is no component of angular velocity on the other principal axes.

To get an explicit expression for the potential energy we must write the direction cosines in terms of and f: = cos _a = - cos ( - f), ß = cos _b = sin ( - f), and = cos _c = 0 because the axis is perpendicular to the orbit plane. The potential energy is then

Since we are assuming that the orbit is given, we need keep only terms that depend on . Expanding the squares of the cosine and the sine in terms of the double angles and dropping all the terms that do not depend on , we find the potential energy for the orientation¹⁹

A Lagrangian for the model spin-orbit coupling problem is then L = T - V:

We introduce the dimensionless ``out-of-roundness'' parameter

and use the fact that the orbit frequency n satisfies Kepler's third law n² a³ = G(M + M'), which is approximately n² a³ = GM' for a small body in orbit around a much more massive one (M << M'). In terms of and n the spin-orbit Lagrangian is

This is a problem with one degree of freedom with terms that vary periodically with time.

The Lagrange equations are derived in the usual manner:

The equation of motion is very similar to that of the periodically driven pendulum. The main difference here is that not only is the strength of the acceleration changing periodically, but in the spin-orbit problem the center of attraction is also varying periodically.

We can give a physical interpretation of this equation of motion. It states that the rate of change of the angular momentum is equal to the applied torque. The torque on the body arises because the body is out of round and the gravitational force varies as the inverse square of the distance. Thus the force per unit mass on the near side of the body is a little more than the acceleration of the body as a whole, and the force per unit mass on the far side of the body is a little less than the acceleration of the body as a whole. Thus, relative to the acceleration of the body as a whole, the far side is forced outward while the inner part of the body is forced inward. The net effect is a torque on the body that tries to align the long axis of the body with the line to the external point mass. If is a bit larger than f then there is a negative torque, and if is a bit smaller than f then there is a positive torque, both of which would align the long axis with the planet if given a fair chance. The torque arises because of the difference of the inverse R² force across the body, so the torque is proportional to R^-3. There is a torque only if the body is out of round, for otherwise there is no handle to pull on. This is reflected in the factor B - A in the expression for the torque. The potential depends only on the moment of inertia, and thus the body has the same dynamics if it is rotated by 180^o. The factor of 2 in the argument of sine reflects this symmetry. This torque is called the ``gravity gradient torque.''

To compute the evolution requires a lot of detailed preparation similar to what has been done for other problems. There are many interesting phenomena to explore. We can take parameters appropriate for the Moon and find that Mr. Moon does not constantly point the same face to the Earth, but instead constantly shakes his head in dismay at what goes on here. If we nudge the Moon a bit, say by hitting it with an asteroid, we find that the long axis oscillates back and forth with respect to the direction that points to the Earth. For the Moon, the orbital eccentricity is currently about 0.05, and the out-of-roundness parameter is about = 0.026. Figure 2.11 shows the angle - f as a function of time for two different values of the ``lunar'' eccentricity. The plot spans 50 lunar orbits, or a little under four years. This Moon has been kicked by a large asteroid and has initial rotational angular velocity equal to 1.01 times the orbit frequency. The initial orientation is = 0. The smooth trace shows the evolution if the orbital eccentricity is set to zero. We see an oscillation with a period of about 40 lunar orbits or about three years. The more wiggly trace shows the evolution of - f with an orbital eccentricity of 0.05, near the current lunar eccentricity. The lunar eccentricity superimposes an apparent shaking of the face of the Moon back and forth with the period of the lunar orbit. Though the Moon does slightly change its rate of rotation during the course of its orbit, most of this shaking is due to the nonuniform motion of the Moon in its elliptical orbit. This oscillation, called the ``optical libration of the Moon,'' allows us to see a bit more than half of the Moon's surface. The longer-period oscillation induced by the kick is called the ``free libration of the Moon.'' It is ``free'' because we are free to excite it by choosing appropriate initial conditions. The mismatch of the orientation of the Moon caused by the optical libration actually produces a periodic torque on the Moon, which slightly speeds it up and slows it down during every orbit. The resulting oscillation is called the ``forced libration of the Moon,'' but it is too small to see in this plot.

The oscillation period of the free libration is easily calculated. We see that the eccentricity of the orbit does not substantially affect the period, so we consider the special case of zero eccentricity. In this case R = a, a constant, and f(t) = n t, where n is the orbital frequency (traditionally called the mean motion). The equation of motion becomes

Let (t) = (t) - n t, and consequently D(t) = D(t) - n, and D² = D². Substituting these, the equation governing the evolution of is

For small deviations from synchronous rotation (small ) this is

so we see that the small-amplitude oscillation frequency of is n. For the Moon, is about 0.026, so the period is about 1/0.026 orbit periods or about 40 lunar orbit periods, which is what we observed.

It is perhaps more fun to see what happens if the out-of-roundness parameter is large. After our experience with the driven pendulum it is no surprise that we find abundant chaos in the spin-orbit problem when the system is strongly driven by having large and significant e. There is indeed one body in the solar system that exhibits chaotic rotation -- Hyperion, a small satellite of Saturn. Though our toy model is not adequate for a complete account of Hyperion, we can show that it exhibits chaotic behavior for parameters appropriate for Hyperion. We take = 0.89 and e = 0.1. Figure 2.12 shows - f for 50 orbits, starting with = 0 and = 1.05. We see that sometimes one face of the body oscillates facing the planet, sometimes the other face oscillates facing the planet, and sometimes the body rotates relative to the planet in either direction.

If we relax our restriction that the spin axis be fixed perpendicular to the orbit, then we find that the Moon maintains this orientation of the spin axis even if nudged a bit, but for Hyperion the spin axis almost immediately falls away from this configuration. The state in which Hyperion on average points one face to Saturn is dynamically unstable to chaotic tumbling. Observations of Hyperion have confirmed that it is chaotically tumbling.

¹⁵ The Legendre polynomials P_l may be obtained by expanding the expression (1 + y² - 2 y x)^-1/2 as a power series in y. The coefficient of y^l is P_l(x). The first few Legendre polynomials are: P₀(x) = 1, P₁(x) = x, P₂(x) = (3/2) x² - (1/2), and so on. The rest satisfy the recurrence relation

¹⁶ This approximate representation of the potential energy is sometimes called MacCullagh's formula.

¹⁷ Watch out, we just reused . It was also used as the constituent index.

¹⁸ Traditionally, the point in the orbit at which the two bodies are closest is called the pericenter and the angle f is called the true anomaly.

¹⁹ The given potential energy differs from the actual potential energy in that non-constant terms that do not depend on and consequently do not affect the evolution of have been dropped.